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The Dimer Model on a Planar Domain: From random domino tilings to limit shapes

by
Alex Bloemendal
|
University of Toronto
Time: 13:00 (Wednesday, Mar. 10, 2010)Location: BA6183, Bahen Center, 40 St George StAbstract:
A dimer covering of a graph, also called a perfect matching, is a subset of edges that covers every vertex exactly once; for an example, tile a chessboard with 2x1 dominoes. Starting with a nice domain in the plane, draw a grid and approximate the domain by a lattice region. The resulting graph might have many dimer coverings; pick one at random. Now using finer and finer grids, a continuum "limit shape" emerges. It is a deterministic object and can be characterized variationally, as a certain "maximal entropy" surface; its principal feature is a closed curve that separates a "frozen" outer region, where all the dimers are aligned, from a "temperate" inner region of disorder. The state-of-the-art description of this picture involves PDE, real algebraic geometry and ergodic theory as well as probability. Background in these subjects will not be assumed, and there will be pretty pictures.